Theorem madebdayim 28307

Description: If a surreal is a member of a made set, its birthday is less than or equal to the level. (Contributed by Scott Fenton, 10-Aug-2024.)

Assertion

Ref	Expression
madebdayim	⊢ (𝑋 ∈ (M‘𝐴) → (bday‘𝑋) ⊆ 𝐴)

Proof of Theorem madebdayim

Dummy variables 𝑎 𝑏 𝑥 𝑦 𝑧 𝑙 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvdm 6945	. . 3 ⊢ (𝑋 ∈ (M‘𝐴) → 𝐴 ∈ dom M)
2		madef 28255	. . . 4 ⊢ M:On⟶𝒫 No
3	2	fdmi 6746	. . 3 ⊢ dom M = On
4	1, 3	eleqtrdi 2874	. 2 ⊢ (𝑋 ∈ (M‘𝐴) → 𝐴 ∈ On)
5		fveq2 6911	. . . . . 6 ⊢ (𝑎 = 𝑏 → (M‘𝑎) = (M‘𝑏))
6		sseq2 3939	. . . . . 6 ⊢ (𝑎 = 𝑏 → ((bday‘𝑥) ⊆ 𝑎 ↔ (bday‘𝑥) ⊆ 𝑏))
7	5, 6	raleqbidv 3338	. . . . 5 ⊢ (𝑎 = 𝑏 → (∀𝑥 ∈ (M‘𝑎)(bday‘𝑥) ⊆ 𝑎 ↔ ∀𝑥 ∈ (M‘𝑏)(bday‘𝑥) ⊆ 𝑏))
8		fveq2 6911	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (bday‘𝑥) = (bday‘𝑦))
9	8	sseq1d 3944	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((bday‘𝑥) ⊆ 𝑏 ↔ (bday‘𝑦) ⊆ 𝑏))
10	9	cbvralvw 3244	. . . . 5 ⊢ (∀𝑥 ∈ (M‘𝑏)(bday‘𝑥) ⊆ 𝑏 ↔ ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏)
11	7, 10	bitrdi 290	. . . 4 ⊢ (𝑎 = 𝑏 → (∀𝑥 ∈ (M‘𝑎)(bday‘𝑥) ⊆ 𝑎 ↔ ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏))
12		fveq2 6911	. . . . 5 ⊢ (𝑎 = 𝐴 → (M‘𝑎) = (M‘𝐴))
13		sseq2 3939	. . . . 5 ⊢ (𝑎 = 𝐴 → ((bday‘𝑥) ⊆ 𝑎 ↔ (bday‘𝑥) ⊆ 𝐴))
14	12, 13	raleqbidv 3338	. . . 4 ⊢ (𝑎 = 𝐴 → (∀𝑥 ∈ (M‘𝑎)(bday‘𝑥) ⊆ 𝑎 ↔ ∀𝑥 ∈ (M‘𝐴)(bday‘𝑥) ⊆ 𝐴))
15		elmade2 28277	. . . . . . . 8 ⊢ (𝑎 ∈ On → (𝑥 ∈ (M‘𝑎) ↔ ∃𝑙 ∈ 𝒫 (O‘𝑎)∃𝑟 ∈ 𝒫 (O‘𝑎)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)))
16	15	adantr 486	. . . . . . 7 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏) → (𝑥 ∈ (M‘𝑎) ↔ ∃𝑙 ∈ 𝒫 (O‘𝑎)∃𝑟 ∈ 𝒫 (O‘𝑎)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)))
17		elpwi 4567	. . . . . . . . . . 11 ⊢ (𝑙 ∈ 𝒫 (O‘𝑎) → 𝑙 ⊆ (O‘𝑎))
18		elpwi 4567	. . . . . . . . . . 11 ⊢ (𝑟 ∈ 𝒫 (O‘𝑎) → 𝑟 ⊆ (O‘𝑎))
19	17, 18	anim12i 625	. . . . . . . . . 10 ⊢ ((𝑙 ∈ 𝒫 (O‘𝑎) ∧ 𝑟 ∈ 𝒫 (O‘𝑎)) → (𝑙 ⊆ (O‘𝑎) ∧ 𝑟 ⊆ (O‘𝑎)))
20		unss 4191	. . . . . . . . . 10 ⊢ ((𝑙 ⊆ (O‘𝑎) ∧ 𝑟 ⊆ (O‘𝑎)) ↔ (𝑙 ∪ 𝑟) ⊆ (O‘𝑎))
21	19, 20	sylib 221	. . . . . . . . 9 ⊢ ((𝑙 ∈ 𝒫 (O‘𝑎) ∧ 𝑟 ∈ 𝒫 (O‘𝑎)) → (𝑙 ∪ 𝑟) ⊆ (O‘𝑎))
22		simpr 490	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ (O‘𝑎)) ∧ 𝑙 <<s 𝑟) → 𝑙 <<s 𝑟)
23		simplll 787	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ (O‘𝑎)) ∧ 𝑙 <<s 𝑟) → 𝑎 ∈ On)
24		dfss3 3902	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑙 ∪ 𝑟) ⊆ (O‘𝑎) ↔ ∀𝑧 ∈ (𝑙 ∪ 𝑟)𝑧 ∈ (O‘𝑎))
25		fveq2 6911	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = 𝑧 → (bday‘𝑦) = (bday‘𝑧))
26	25	sseq1d 3944	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = 𝑧 → ((bday‘𝑦) ⊆ 𝑏 ↔ (bday‘𝑧) ⊆ 𝑏))
27	26	rspccv 3577	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏 → (𝑧 ∈ (M‘𝑏) → (bday‘𝑧) ⊆ 𝑏))
28	27	ralimi 3103	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏 → ∀𝑏 ∈ 𝑎 (𝑧 ∈ (M‘𝑏) → (bday‘𝑧) ⊆ 𝑏))
29		rexim 3107	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑏 ∈ 𝑎 (𝑧 ∈ (M‘𝑏) → (bday‘𝑧) ⊆ 𝑏) → (∃𝑏 ∈ 𝑎 𝑧 ∈ (M‘𝑏) → ∃𝑏 ∈ 𝑎 (bday‘𝑧) ⊆ 𝑏))
30	28, 29	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏 → (∃𝑏 ∈ 𝑎 𝑧 ∈ (M‘𝑏) → ∃𝑏 ∈ 𝑎 (bday‘𝑧) ⊆ 𝑏))
31	30	adantl 487	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏) → (∃𝑏 ∈ 𝑎 𝑧 ∈ (M‘𝑏) → ∃𝑏 ∈ 𝑎 (bday‘𝑧) ⊆ 𝑏))
32		elold 28278	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ On → (𝑧 ∈ (O‘𝑎) ↔ ∃𝑏 ∈ 𝑎 𝑧 ∈ (M‘𝑏)))
33	32	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏) → (𝑧 ∈ (O‘𝑎) ↔ ∃𝑏 ∈ 𝑎 𝑧 ∈ (M‘𝑏)))
34		bdayon 28171	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (bday‘𝑧) ∈ On
35		onelssex 6433	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((bday‘𝑧) ∈ On ∧ 𝑎 ∈ On) → ((bday‘𝑧) ∈ 𝑎 ↔ ∃𝑏 ∈ 𝑎 (bday‘𝑧) ⊆ 𝑏))
36	34, 35	mpan 703	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ On → ((bday‘𝑧) ∈ 𝑎 ↔ ∃𝑏 ∈ 𝑎 (bday‘𝑧) ⊆ 𝑏))
37	36	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏) → ((bday‘𝑧) ∈ 𝑎 ↔ ∃𝑏 ∈ 𝑎 (bday‘𝑧) ⊆ 𝑏))
38	31, 33, 37	3imtr4d 297	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏) → (𝑧 ∈ (O‘𝑎) → (bday‘𝑧) ∈ 𝑎))
39	38	ralimdv 3180	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏) → (∀𝑧 ∈ (𝑙 ∪ 𝑟)𝑧 ∈ (O‘𝑎) → ∀𝑧 ∈ (𝑙 ∪ 𝑟)(bday‘𝑧) ∈ 𝑎))
40	24, 39	biimtrid 245	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏) → ((𝑙 ∪ 𝑟) ⊆ (O‘𝑎) → ∀𝑧 ∈ (𝑙 ∪ 𝑟)(bday‘𝑧) ∈ 𝑎))
41	40	imp 412	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ (O‘𝑎)) → ∀𝑧 ∈ (𝑙 ∪ 𝑟)(bday‘𝑧) ∈ 𝑎)
42	41	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ (O‘𝑎)) ∧ 𝑙 <<s 𝑟) → ∀𝑧 ∈ (𝑙 ∪ 𝑟)(bday‘𝑧) ∈ 𝑎)
43		bdayfun 28166	. . . . . . . . . . . . . . . . 17 ⊢ Fun bday
44		oldssno 28260	. . . . . . . . . . . . . . . . . . 19 ⊢ (O‘𝑎) ⊆ No
45		sstr 3921	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑙 ∪ 𝑟) ⊆ (O‘𝑎) ∧ (O‘𝑎) ⊆ No) → (𝑙 ∪ 𝑟) ⊆ No)
46	44, 45	mpan2 704	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑙 ∪ 𝑟) ⊆ (O‘𝑎) → (𝑙 ∪ 𝑟) ⊆ No)
47		bdaydm 28168	. . . . . . . . . . . . . . . . . 18 ⊢ dom bday = No
48	46, 47	sseqtrrdi 3954	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑙 ∪ 𝑟) ⊆ (O‘𝑎) → (𝑙 ∪ 𝑟) ⊆ dom bday)
49		funimass4 6975	. . . . . . . . . . . . . . . . 17 ⊢ ((Fun bday ∧ (𝑙 ∪ 𝑟) ⊆ dom bday) → ((bday “ (𝑙 ∪ 𝑟)) ⊆ 𝑎 ↔ ∀𝑧 ∈ (𝑙 ∪ 𝑟)(bday‘𝑧) ∈ 𝑎))
50	43, 48, 49	sylancr 599	. . . . . . . . . . . . . . . 16 ⊢ ((𝑙 ∪ 𝑟) ⊆ (O‘𝑎) → ((bday “ (𝑙 ∪ 𝑟)) ⊆ 𝑎 ↔ ∀𝑧 ∈ (𝑙 ∪ 𝑟)(bday‘𝑧) ∈ 𝑎))
51	50	adantl 487	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ (O‘𝑎)) → ((bday “ (𝑙 ∪ 𝑟)) ⊆ 𝑎 ↔ ∀𝑧 ∈ (𝑙 ∪ 𝑟)(bday‘𝑧) ∈ 𝑎))
52	51	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ (O‘𝑎)) ∧ 𝑙 <<s 𝑟) → ((bday “ (𝑙 ∪ 𝑟)) ⊆ 𝑎 ↔ ∀𝑧 ∈ (𝑙 ∪ 𝑟)(bday‘𝑧) ∈ 𝑎))
53	42, 52	mpbird 260	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ (O‘𝑎)) ∧ 𝑙 <<s 𝑟) → (bday “ (𝑙 ∪ 𝑟)) ⊆ 𝑎)
54		cutbdaybnd 28214	. . . . . . . . . . . . 13 ⊢ ((𝑙 <<s 𝑟 ∧ 𝑎 ∈ On ∧ (bday “ (𝑙 ∪ 𝑟)) ⊆ 𝑎) → (bday‘(𝑙 |s 𝑟)) ⊆ 𝑎)
55	22, 23, 53, 54	syl3anc 1399	. . . . . . . . . . . 12 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ (O‘𝑎)) ∧ 𝑙 <<s 𝑟) → (bday‘(𝑙 |s 𝑟)) ⊆ 𝑎)
56		fveq2 6911	. . . . . . . . . . . . 13 ⊢ ((𝑙 |s 𝑟) = 𝑥 → (bday‘(𝑙 |s 𝑟)) = (bday‘𝑥))
57	56	sseq1d 3944	. . . . . . . . . . . 12 ⊢ ((𝑙 |s 𝑟) = 𝑥 → ((bday‘(𝑙 |s 𝑟)) ⊆ 𝑎 ↔ (bday‘𝑥) ⊆ 𝑎))
58	55, 57	syl5ibcom 248	. . . . . . . . . . 11 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ (O‘𝑎)) ∧ 𝑙 <<s 𝑟) → ((𝑙 |s 𝑟) = 𝑥 → (bday‘𝑥) ⊆ 𝑎))
59	58	expimpd 459	. . . . . . . . . 10 ⊢ (((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ (O‘𝑎)) → ((𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) → (bday‘𝑥) ⊆ 𝑎))
60	59	ex 418	. . . . . . . . 9 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏) → ((𝑙 ∪ 𝑟) ⊆ (O‘𝑎) → ((𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) → (bday‘𝑥) ⊆ 𝑎)))
61	21, 60	syl5 35	. . . . . . . 8 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏) → ((𝑙 ∈ 𝒫 (O‘𝑎) ∧ 𝑟 ∈ 𝒫 (O‘𝑎)) → ((𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) → (bday‘𝑥) ⊆ 𝑎)))
62	61	rexlimdvv 3222	. . . . . . 7 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏) → (∃𝑙 ∈ 𝒫 (O‘𝑎)∃𝑟 ∈ 𝒫 (O‘𝑎)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) → (bday‘𝑥) ⊆ 𝑎))
63	16, 62	sylbid 243	. . . . . 6 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏) → (𝑥 ∈ (M‘𝑎) → (bday‘𝑥) ⊆ 𝑎))
64	63	ralrimiv 3157	. . . . 5 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏) → ∀𝑥 ∈ (M‘𝑎)(bday‘𝑥) ⊆ 𝑎)
65	64	ex 418	. . . 4 ⊢ (𝑎 ∈ On → (∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏 → ∀𝑥 ∈ (M‘𝑎)(bday‘𝑥) ⊆ 𝑎))
66	11, 14, 65	tfis3 7896	. . 3 ⊢ (𝐴 ∈ On → ∀𝑥 ∈ (M‘𝐴)(bday‘𝑥) ⊆ 𝐴)
67		fveq2 6911	. . . . 5 ⊢ (𝑥 = 𝑋 → (bday‘𝑥) = (bday‘𝑋))
68	67	sseq1d 3944	. . . 4 ⊢ (𝑥 = 𝑋 → ((bday‘𝑥) ⊆ 𝐴 ↔ (bday‘𝑋) ⊆ 𝐴))
69	68	rspccv 3577	. . 3 ⊢ (∀𝑥 ∈ (M‘𝐴)(bday‘𝑥) ⊆ 𝐴 → (𝑋 ∈ (M‘𝐴) → (bday‘𝑋) ⊆ 𝐴))
70	66, 69	syl 18	. 2 ⊢ (𝐴 ∈ On → (𝑋 ∈ (M‘𝐴) → (bday‘𝑋) ⊆ 𝐴))
71	4, 70	mpcom 39	1 ⊢ (𝑋 ∈ (M‘𝐴) → (bday‘𝑋) ⊆ 𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900   ∪ cun 4152  𝒫 cpw 4560   class class class wbr 5106  dom cdm 5656   “ cima 5659  Oncon0 6383  Fun wfun 6555  ‘cfv 6561  (class class class)co 7446  Nocsur 28030  bdaycbday 28032   <<s cslts 28176   |s ccuts 28178  Mcmade 28241  Ocold 28242

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-1o 8507  df-2o 8508  df-no 28033  df-lts 28034  df-bday 28035  df-slts 28177  df-cuts 28179  df-made 28246  df-old 28247

This theorem is used by:  oldbdayim  28308  madebday  28319  onsbnd  28700  onsbnd2  28701